*An existence theorem for conformally-flat metrics*

**Abstract:**
The goal of this talk is to prove a generalization of Cliff Taubes' theorem on
existence of self-dual Riemannian metrics on connected sums of a given smooth
oriented 4-manifolds with a large number of CP^2's.

Recall that a smooth n-manifold M is called almost parallelizable if (M minus point) has trivial tangent bundle. For instance, if n=4 then almost parallelizable is equivalent to Spin, if n=3 then it is equivalent to orientable.